Skip to main content

Estimation with multivariate outcomes having nonignorable item nonresponse


To estimate unknown population parameters based on \({\varvec{y}}\), a vector of multivariate outcomes having nonignorable item nonresponse that directly depends on \({\varvec{y}}\), we propose an innovative inverse propensity weighting approach when the joint distribution of \({\varvec{y}}\) and associated covariate \({\varvec{x}}\) is nonparametric and the nonresponse probability conditional on \({\varvec{y}}\) and \({\varvec{x}}\) has a parametric form. To deal with the identifiability issue, we utilize a nonresponse instrument \({\varvec{z}}\), an auxiliary variable related to \({\varvec{y}}\) but not related to the nonresponse probability conditional on \({\varvec{y}}\) and \({\varvec{x}}\). We utilize a modified generalized method of moments to obtain estimators of the parameters in the nonresponse probability. Simulation results are presented and an application is illustrated in a real data set.

This is a preview of subscription content, access via your institution.


  • Cho, H., Hong, H. G., Kim, M. O. (2016). Efficient quantile marginal regression for longitudinal data with dropouts. Biostatistics, 17, 561–575.

    MathSciNet  Article  Google Scholar 

  • Greenlees, J. S., Reece, W. S., Zieschang, K. D. (1982). Imputation of missing values when the probability of response depends on the variable being imputed. Journal of the American Statistical Association, 77, 251–261.

    Article  Google Scholar 

  • Hall, A. R. (2005). Generalized method of moments. New York: Oxford University Press.

    MATH  Google Scholar 

  • Hansen, L. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50, 1029–1054.

    MathSciNet  Article  Google Scholar 

  • Li, S., Shao, J. (2022). Nonignorable item nonresponse in panel data. Statistical Theory and Related Fields, 6, 58–71.

    MathSciNet  Article  Google Scholar 

  • Little, R. J. A., Rubin, D. B. (2002). Statistical analysis with missing data (2nd ed.). New York: Wiley.

    Book  Google Scholar 

  • Robins, J. M., Rotiv, Y. (1997). Toward a curse of dimensionality appropriate (CODA) asymptotic theory for semiparametric models. Statistics in Medicine, 16, 285–319.

    Article  Google Scholar 

  • Rubin, D. B. (1976). Inference with missing data. Biometrika, 63, 581–592.

    MathSciNet  Article  Google Scholar 

  • Shao, J., Tu, D. (1995). The jackknife and bootstrap. New York: Springer-Verlag.

    Book  Google Scholar 

  • Shao, J., Wang, L. (2016). Semiparametric inverse propensity weighting for nonignorable missing data. Biometrika, 103, 175–187.

    MathSciNet  Article  Google Scholar 

  • Shao, J., Zhang, J. (2015). A transformation approach in linear mixed-effect models with informative missing responses. Biometrika, 102, 107–119.

    MathSciNet  Article  Google Scholar 

  • Tang, G., Little, R. J. A., Raghunathan, T. E. (2003). Analysis of multivariate missing data with nonignorable nonresponse. Biometrika, 90, 747–764.

    MathSciNet  Article  Google Scholar 

  • Wang, S., Shao, J., Kim, J. K. (2014). An instrumental variable approach for identification and estimation with nonignorable nonresponse. Statistica Sinica, 24, 1097–1116.

    MathSciNet  MATH  Google Scholar 

  • Wu, M. C., Carroll, R. J. (1988). Estimation and comparison of changes in the presence of informative right censoring by modeling the censoring process. Biometrics, 44, 175–188.

    MathSciNet  Article  Google Scholar 

  • Xu, L., Shao, J. (2009). Estimation in longitudinal or panel data models with random-effect-based missing responses. Biometrics, 65, 1175–1183.

    MathSciNet  Article  Google Scholar 

  • Yuan, Y., Yin, G. (2010). Bayesian quantile regression for longitudinal studies with nonignorable missing data. Biometrics, 66, 105–114.

    MathSciNet  Article  Google Scholar 

  • Zhao, J., Shao, J. (2015). Semiparametric pseudo likelihoods in generalized linear models with nonignorable missing data. Journal of American Statistical Association, 110, 1577–1590.

    MathSciNet  Article  Google Scholar 

Download references


We are grateful to the associate editor and two referees for comments and suggestions that led to improvements of the paper. Lyu Ni’s research was supported by the Shanghai Sailing Program 22YF1411300. Jun Shao’s research was supported by the National Natural Science Foundation of China Grant 11831008 and the U.S. National Science Foundation Grant DMS-1914411.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Jun Shao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ni, L., Shao, J. Estimation with multivariate outcomes having nonignorable item nonresponse. Ann Inst Stat Math (2022).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI:


  • Generalized method of moments
  • Item nonresponse
  • Inverse propensity weighting
  • Multivariate outcome
  • Nonresponse instrument